simon-5502-12-slides

Topics to be covered

What you will learn
- Mathematical formulation of random intercepts model
- Description of HIV-intervention data
- Random intercepts model using hiv-intervention data
- Mathematical formulation of random slopes model
- Assumptions and complications
- Sample size justification

Longitudinal data

Measurements taken at different times
- Emphasis in changes over time

In the previous module, I talked about hierarchical models and mentioned a particular case, longitudinal data, that I want to talk more about in this presentation.

Longitudinal data is similar to repeated measures data. With both, you measure the same subject repeatedly. With longitudinal data, often the emphasis is in changes that occur over time. Repeated measurements, in contrast, emphasize different treatments with the hope that the time gaps between the measurements are small enough that you don’t see changes over time.

The differences between longitudinal data, repeated measures data, or hierarchical data are subtle. Perhaps these are distinctions without a difference. I decided to separate out longitudinal data for a different module perhaps more out of the desire to split a complex topic into smaller bite-sized pieces.

Random intercepts model, 1

Simplest pattern for longitudinal data
\(Y_{ij},\ i=1,...,n;\ j=1,...,k\)
- n subjects, k time points
\(t_j\), time of jth measurement
- First time is often zero

Random intercepts model, 2

\(Y_{ij}=\beta_0+u_{0i}+\beta_1 t_j + \epsilon_{ij}\)
- \(\beta_0\) and \(\beta_1\) are unknown constants
- \(u_{0i}\) and \(\epsilon_{ij}\) are normally distributed
  - \(SD(u_{0i})=\sigma_{intercept}\)
  - \(SD(\epsilon_{ij})=\sigma_{error}\)

Random intercepts model, 3

\(SD(Y_{ij})=\sqrt{\sigma^2_{intercept}\ +\ \sigma^2_{error}}\)
\(Corr(Y_{ij}, Y_{im})=\frac{\sigma^2_{intercept}}{\sigma^2_{intercept}\ +\ \sigma^2_{error}}\)

Random intercepts illustrated, 1

Random intercepts illustrated, 2

Illustration of random intercepts with real data

Break #1

What you have learned
- Mathematical formulation of random intercepts model
What’s coming next
- Description of HIV-intervention data

Description of hiv-intervention data, 1

data_dictionary: hiv-intervention.txt

source: OzDASL website

description: |
  This is a longitudinal study of an intervention in 14-18 adolescents  intended to increase the frequency of condom protected sex. Subjects  were allocated randomly to treatment or control groups. All were evaluated prior to the intervention, immediately  after the intervention, 6 months and  12 months after the intervention.The outcome variable is the logarithm-transformed frequency of condom-protected sex ( log(Y+1) )."

Description of hiv-intervention data, 2

BST:
  label: treatment group
  values:
    '1': BST intervention
    '0': control
Pre:
  label: Log-frequency of protected sex before the intervention
Post:
  label: Log-frequency of protected sex after the intervention
FU6:
  label: Log-frequency of protected sex reported at the 6 months follow-up
FU12:
  label: Log-frequency of protected sex reported at the 12 months follow-up

Glimpse of the hiv-intervention data

Rows: 20
Columns: 5
$ BST  <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
$ Pre  <dbl> 7, 25, 50, 16, 33, 10, 13, 22, 4, 17, 0, 69, 5, 4, 35, 7, 51, 25,…
$ Post <dbl> 22, 10, 36, 38, 25, 7, 33, 20, 0, 16, 0, 56, 0, 24, 8, 0, 53, 0, …
$ FU6  <dbl> 13, 17, 49, 34, 24, 23, 27, 21, 12, 20, 0, 14, 0, 0, 0, 9, 8, 0, …
$ FU12 <dbl> 14, 24, 23, 24, 25, 26, 24, 11, 0, 10, 0, 36, 5, 0, 0, 37, 26, 15…

Plot of the data, 1

Plot of the data, 2

Glimpse of the restructured and simplified data

Rows: 30
Columns: 4
$ BST           <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,…
$ id            <int> 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7,…
$ t             <dbl> 0, 6, 12, 0, 6, 12, 0, 6, 12, 0, 6, 12, 0, 6, 12, 0, 6, …
$ protected_sex <dbl> 22, 13, 14, 10, 17, 24, 36, 49, 23, 38, 34, 24, 25, 24, …

Break #2

What you have learned
- Description of HIV-intervention data
What’s coming next
- Random intercepts model using hiv-intervention data

Random intercepts analysis, 1

Linear mixed model fit by REML ['lmerMod']
Formula: protected_sex ~ t + (1 | id)
   Data: hiv_3

REML criterion at convergence: 215.1

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-1.8385 -0.6028  0.0364  0.5183  2.1998

Random intercepts analysis, 2

Random effects:
 Groups   Name        Variance Std.Dev.
 id       (Intercept) 69.57    8.341   
 Residual             53.34    7.304   
Number of obs: 30, groups:  id, 10

Random intercepts analysis, 3

Fixed effects:
            Estimate Std. Error t value
(Intercept)  22.2333     3.3768   6.584
t            -0.2167     0.2722  -0.796

Random intercepts analysis, 4

Correlation of Fixed Effects:
  (Intr)
t -0.484

Live demo, fitting a random intercepts model

Break #3

What you have learned
- Random intercepts model using hiv-intervention data
What’s coming next
- Mathematical formulation of random slopes model

Random slopes model, 1

Same notation for the time and outcome variables
\(Y_{ij},\ i=1,...,n;\ j=1,...,k\)
- n subjects, k time points
\(t_j\), time of jth measurement

Random slopes model, 2

\(Y_{ij}=\beta_0+u_{0i}+\beta_1 t_j+u_{1i} t_j+\epsilon_{ij}\)
- \(\beta_0\) and \(\beta_1\) are unknown constants
- \(u_{0i}\), \(u_{1i}\), and \(\epsilon_{ij}\) are normally distributed
  - \(SD(u_{0i})=\sigma_{intercept}\)
  - \(SD(u_{1i})=\sigma_{slope}\)
  - \(SD(\epsilon_{ij})=\sigma_{error}\)

Random slopes illustrated, 1

Random slopes illustrated, 2

Break #4

What you have learned
- Mathematical formulation of random slopes model
What’s coming next
- Assumptions and complications

Assumptions

Independence
- Only between subjects
Normality
- Residuals
- Random intercepts and/or slopes
Linearity

Linearity check

Normality check within clusters

Normality check for random intercepts

Complications

Not a problem
- Missing values
- Better than Last Observation Carried Forward
Problems (more tedious than difficult)
- Interactions
- Nonlinear trends
- Covariates
  - Between patients
  - Within patients

Missing values are normally a big headache, but less so for the random intercepts or slopes models. If a patient missed a visit at a particular time point, you can easily extrapolate from the visits before and after. You are assuming linearity, after all.

In some clinical trials, if a patient came at the intermediate evaluation but did not show up at the final evaluation, the research team would replace the missing final outcome with the intermediate outcome. This technique, called Last Observation Carried Forward (LOCF) was not very popular when it was introduced and has been pretty much discredited. The random intercepts or slopes model would extrapolate the trend from the intermediate value, which works much better.

Interactions are always difficult, and when the interactions involve time, it can get a bit messy. Nonlinear trends over time are also a bit of a problem. In both cases, though, the work is more tedious than difficult. Interactions and nonlinearity mean that your interpretation of the results will require a bit more thought and you can’t come up with as simple a story to tell.

A covariate is a variable which is not of direct interest in the research, but one that you must take account of in order to produce a credible analysis. In just about any cancer study, you should track whether the patient is a smoker. It’s not something you’re interested in testing. The role of smoking in lung cancer and most other types of cancer was established many decades ago. You still have to account for smoking though because it can explain so much of the variation in your outcome. Failure to account for smoking would greatly reduce your power and precision.

There are two types of covariates. The ones that are fixed and do not change over time are called time constant covariates or between subject covariates. Patient demographics are time constant. Measurements done at baseline to assess how ill the patient was at the start of the study are time constant.

Covariates that change over time are called time varying covariates or within subject covariates. The extent to which a patient complies with taking his/her medication is a time varying covariate. Seasonal changes in temperature, humidity, or pollen counts are time varying covariates.

There is one important distinction between time constant and time varying covariates. The latter are much better at removing variation from your outcome, and can greatly improve your power and precision.

Live demo, checking assumptions

Break #5

What you have learned
- Assumptions and complications
What’s coming next
- Sample size justification

Effect of co-housing on sample size calculations, 1

Read a bit from this abstract:

“Analysis of preclinical lifespan studies often assume that outcome data from co-housed animals are independent. In practice, treatments, such as controlled feeding or putative life-extending compounds, are applied to whole housing units, and as a result the outcomes are potentially correlated within housing units. We consider intra-class (here, intra-cage) correlation in three published and two unpublished lifespan studies of aged mice encompassing more than 20 thousand observations. We show that the independence assumption underlying common analytic techniques does not hold in these data, particularly for traits associated with frailty. We describe and demonstrate various analytical tools available to accommodate this study design and highlight a limitation of standard variance components models (i.e., linear mixed models) which are the usual statistical tool for handling correlated errors. Through simulations, we examine the statistical biases resulting from intra-cage correlations with similar magnitudes as observed in these case studies and discuss implications for power and reproducibility.”

Effect of co-housing on sample size calculations, 2

Effect of co-housing on sample size calculations, 3

Effect of co-housing on sample size calculations, 4

Sample size estimate without clustering

delta <- 4
sd_independence <- 11.1

power.t.test(
    n=NULL, 
    delta=4,
    sd=sd_independence,
    sig.level=0.05,
    power=0.8,
    type="two.sample") |> 
    tidy() -> sample_size_1

sample_size_1

# A tibble: 1 × 5
      n delta    sd sig.level power
  <dbl> <dbl> <dbl>     <dbl> <dbl>
1  122.     4  11.1      0.05   0.8

Sample size estimate with clustering, 4 animals per cage

deff <- 1+(4-1)*0.049
sd_correlated <- sd_independence * sqrt(deff)

power.t.test(
    n=NULL, 
    delta=4,
    sd=sd_correlated,
    sig.level=0.05,
    power=0.8,
    type="two.sample") |> 
    tidy()

# A tibble: 1 × 5
      n delta    sd sig.level power
  <dbl> <dbl> <dbl>     <dbl> <dbl>
1  140.     4  11.9      0.05   0.8

Sample size estimate with clustering, 8 animals per cage

deff <- 1+(8-1)*0.049
sd_correlated <- sd_independence * sqrt(deff)

power.t.test(
    n=NULL, 
    delta=4,
    sd=sd_correlated,
    sig.level=0.05,
    power=0.8,
    type="two.sample") |> 
    tidy()

# A tibble: 1 × 5
      n delta    sd sig.level power
  <dbl> <dbl> <dbl>     <dbl> <dbl>
1  163.     4  12.9      0.05   0.8

You could also just multiply the sample size by the design effect

sample_size_1$n * (1+(4-1)*0.049)

[1] 139.7624

sample_size_1$n * (1+(8-1)*0.049)

[1] 163.6451

Summary

What you have learned
- Mathematical formulation of random intercepts model
- Description of HIV-intervention data
- Random intercepts model using hiv-intervention data
- Mathematical formulation of random slopes model
- Assumptions and complications
- Sample size justification